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Mirrors > Home > MPE Home > Th. List > dmv | Structured version Visualization version GIF version |
Description: The domain of the universe is the universe. (Contributed by NM, 8-Aug-2003.) |
Ref | Expression |
---|---|
dmv | ⊢ dom V = V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssv 3911 | . 2 ⊢ dom V ⊆ V | |
2 | dmi 5775 | . . 3 ⊢ dom I = V | |
3 | ssv 3911 | . . . 4 ⊢ I ⊆ V | |
4 | dmss 5756 | . . . 4 ⊢ ( I ⊆ V → dom I ⊆ dom V) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ dom I ⊆ dom V |
6 | 2, 5 | eqsstrri 3922 | . 2 ⊢ V ⊆ dom V |
7 | 1, 6 | eqssi 3903 | 1 ⊢ dom V = V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 Vcvv 3398 ⊆ wss 3853 I cid 5439 dom cdm 5536 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-dm 5546 |
This theorem is referenced by: (None) |
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